## Abstract

Having in mind the previously developed models (Chap. 2) and pricing approaches (Chap. 3), this chapter intends to provide the reader with an empirical comparison of them. Even though part of the arguments in this book can seem rather theoretical, the underlying intention behind all of this is to provide a pricing technology that is based on realistic models of financial markets. In this chapter, we will be using a dataset of European options on the S&P500 to test and compare those models. We will not only compare the time series approaches that have been discussed earlier, but compare them to various “standards” of the industry, namely the calibrated and estimated versions of the Heston (1993) model (or at the very least their Heston and Nandi (2000) discrete time counterparts). By doing so, we aim at putting a greater emphasize on a key point most of quantitative analysts around the world are familiar with: rough scales of magnitude for option pricing errors that anyone could expect from a good option pricing model.

### Keywords

- Option Price
- Implied Volatility
- Strike Price
- Underlying Asset
- Price Performance

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

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## Buying options

## Notes

- 1.
- 2.
The value of an option is made of two different pieces. First, the intrinsic value that is the value of an immediate exercise of the option, either 0 or the difference between the current price of the underlying asset and the strike price in the case of a call option. The second piece is the time value, that is the value coming from the fact that the option is not exercised today. Usually, this time value is low.

- 3.
In this case, the historical and the risk neutral volatilities coincide (see Proposition 3.2.2).

- 4.
Computing the derivative of Eq. (4.3) with respect to

*σ*, we obtain the so-called Véga (sensitivity of the price with respect to the volatility) that is strictly positive: the call option price is a strictly increasing (and thus invertible) function of the volatility. Given the integral involved in the formula, there is no closed-form expression to such an inverse function. It has to be numerically approximated. One possibility (that is used in the following) is to minimize the quadratic difference between the theoretical price and the actual market price. We also mention that very accurate approximations for*N*^{−1}exist (see Cody 1969) and are implemented in R via the*qnorm*function of the*stats*package. - 5.
By “series” here, we refer to a dataset of implied volatilities for a given moneyness and a fixed time to maturity, across the dates of our sample.

- 6.
- 7.
N being the number of different (moneyness,time to maturity) pairs in our dataset and T the number of dates.

- 8.
See Eq. (7), page 49 of Cont and da Fonseca (2002).

- 9.
*h*is actually working as a smoothing parameter: the bigger this parameter is and the smoother is the function. An usual way to set the smoothing parameter*h*associated to a dataset*x*(here the moneyness or the maturity values) is to set it to be equal to \(\frac{1.8\sigma _{x}} {n^{2/5}}\), where*n*is the total number of observations in*x*and*σ*_{ x }the corresponding empirical standard deviation. - 10.
Here

*C*(*T*−*t*,*K*) denotes the price of a call option with strike*K*and time to maturity*T*−*t*. - 11.
We have already seen in Sect. 2.7 that the Heston and Nandi model weakly converges under the historical probability toward the Heston diffusion when parametric constraints are well chosen.

- 12.
For the GARCH(1,1) model, the long term variance is given by \(\frac{a_{0}} {1-(a_{1}+b_{1})}\) and the mean-reverting structure of the volatility a consequence of (2.16).

- 13.
In this case, the computation of option prices is not as fast as in the case of closed-form formulas, but it remains much faster than when using Monte Carlo methods.

- 14.
The purpose of this chapter is to provide the reader with a guide on how to use an option pricing model in the standard way: by standard, we mean that before computing option prices, we need to find a set of values for the parameters that fits the distribution of the returns. This step should be an essential one and is often overlooked, as rather liquid option markets made this step useless. When vanilla option markets are liquid, the parameters under the pricing measure can be recovered from option prices—a method usually referred to as “calibration”. However, liquidity traps and the need to price options on new underlying assets make this calibration step inadequate from time to time. What is more, we should mention that the understanding of the economic intuitions hidden behind an option pricing model and the study of historical patterns should make the quantitative analysts around the world more cautious about the ability of pricing models to be reliable under any circumstances.

- 15.
An interested reader should browse the documentation associated with the following optimizers: optim, nlm and nlminb. Each of them provides optimization solutions depending on the nature of the optimization problem the user is faced with.

- 16.
The order of the arguments of the optim function has been changing over the years depending on the version of R. We advise the interested reader to look at the documentation associated with the function by typing ?optim in the software console.

- 17.
- 18.
On this point, see Bates (1997).

- 19.
Roughly speaking, the FFT is an efficient algorithm due to Cooley and Tukey (1965) which allows to compute the sums

*F*_{0},*…*,*F*_{ N−1}where$$\displaystyle{ F_{k} =\sum \limits _{ j=0}^{N-1}e^{\frac{-2i\pi \mathit{jk}} {N} }x(j) }$$(4.21)using only \(\mathrm{O}(\mathit{Nlog}(N))\) operations instead of O(

*N*^{2}). - 20.
In practice we will take

*α*= 2, value that correctly works in our empirical study. See Carr and Madan (1999) for a discussion on the choice of this parameter. - 21.
In R, the complex number i is denoted by 1i.

- 22.
- 23.
The price of an European call option can be decomposed into an intrinsic value and a time value. The intrinsic value at time

*t*is simply equal to (*S*_{ t }−*K*)_{+}, that is what the bearer of such an option could obtain if he could exercise it immediately. The time value is defined as the difference between the current price of the option and the intrinsic value of the option. It represents how much the market values the investor’s patience until the time to maturity, hence its name. - 24.
- 25.
In our empirical results, we use the squared value of the VIX as a measure of the risk neutral variance to determine the value of

*π*. - 26.
We do not study the GJR specification in this section because we have seen in Table 2.4 that it is a special case of the APARCH model.

- 27.
As introduced in Chaps. 2 and 3, we will refer to these mix of variance dynamics and distributions using EGARCH and APARCH for the two volatility structures and GH for the Generalized Hyperbolic distribution, MN for the Mixture of Gaussian distributions and “Jumps” for the jump component. The “EGARCH-Jump” model will therefore be the mixture of an EGARCH variance dynamics with the jump component as a conditional distribution.

- 28.
An interested reader will read with interest Chorro et al. (2014).

- 29.
Say we deal with a time series model for the log-returns whose estimated conditional density at time

*t*is \(f(Y _{t}\vert \underline{Y _{t-1}},\hat{\theta }_{1})\), where \(\underline{Y _{t-1}} = (Y _{1},\ldots,Y _{t-1})\) and \(\hat{\theta }_{1}\) is the vector of parameters describing the shape of this conditional distribution and the volatility structure estimated from methodology 1. We compare this estimation method to another one defined by the conditional density \(f(Y _{t}\vert \underline{Y _{t-1}},\hat{\theta }_{2})\), with \(\hat{\theta }_{2}\) being the estimated parameters obtained from this second method. The null hypothesis of the test is “methods 1 and 2 provide a similar fit of the log-return’s conditional distribution using the same underlying model”. The corresponding test statistic is then:$$\displaystyle{t_{1,2} = \frac{1} {n}\sum _{t=1}^{n}\left (\log f(Y _{t}\vert \underline{Y _{ t-1}},\hat{\theta }_{1}) -\log f(Y _{t}\vert \underline{Y _{t-1}},\hat{\theta }_{2})\right )}$$where

*n*is the total number of observations available. Under the null hypothesis$$\displaystyle{ \frac{t_{1,2}} {\hat{\sigma }_{n}} \sqrt{n}\mathop{ \rightarrow }\limits_{ n \rightarrow +\infty }\mathcal{N}(0, 1), }$$(4.53)where \(\hat{\sigma }_{n}\) is a properly selected estimator for the statistic volatility. Here, as proposed in Amisano and Giacomini (2007), we use a Newey-West estimator, with a lag empirically retained to be large (around 25).

- 30.
- 31.
- 32.
Explicit expressions of the conditional moment generating functions for the distributions considered in this empirical part are given in Sect. 3.4.2.

- 33.
See e.g. Andersen et al. (2001).

- 34.
Those results are still based on the use of the REC estimates.

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Chorro, C., Guégan, D., Ielpo, F. (2015). Empirical Performances of Discrete Time Series Models. In: A Time Series Approach to Option Pricing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45037-6_4

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